General Principles

Abstract

The general mathematical principles of quantum electrodynamics or any other quantum field theory are copied from those of the usual nonrelativistic quantum mechanics of (point) particles. Hence quantum electrodynamics also deals with a state vector in a “Hilbert space ”and with a set of linear operators in this space, the “field operators”. The latter are the dynamical variables of the theory and correspond to the classical fields, in the sense of the correspondence principle. As an example, certain operators correspond to the classical electromagnetic potentials. The four-dimensional space-time coordinates x ₁ = x, x ₂ = y, x ₃ = z,x ₄ = ix ₀ = ict, which we shall often designate simply by x, are not operators in the sense of the nonrelativistic quantum mechanics. Rather, they must be understood as “indices” or “labels” for the field operators. Thus with each point x there are associated a finite number of field operators (eight in quantum electrodynamics), which we shall designate as φ _α (x) in the first chapter. The index α distinguishes the various fields; for example, it distinguishes the four components of the electromagnetic potential and the four Dirac field operators.